I like both but I think Nash suits your last name better and gives the name a handsome and approachable vibe - immediately likeable vibe. Its only downside is the slight “Shroberts” sound when said together - but full names are rarely said.
I like Rome but I don’t find R and R the easiest alliteration - it’s quite easy to swallow a double R name (I know because my brother is a Rhys and our last name is an R name). It can be slurred.
Rome is probably more spunky and fun but Nash is more rugged and handsome to me, and it’ll get less “Rome as in Rome or Roam? Rome as in Romeo? Short for Roman?”. I just think Nash is lower risk and goes with the heritage of your last name better (ie feels more cohesive) and will age really nicely. Who wouldn’t like to meet an adult Nash?
Edit: I almost wonder if there is a halfway point name, like Heath (swoony with Roberts!), or a fave of mine, Jude!